1. Field of Invention
The present invention relates to airfoil actuator feedback controls and, more specifically, to the use of proper orthogonal decomposition (POD) and modified linear stochastic estimation (mLSE) for the determining the flow velocities over an airfoil and correspondingly controlling the actuators of the airfoil.
2. Description of Prior Art
The present invention is based on a foundation laid by Taylor, J. A. and Glauser, M. N., 2002, Towards Practical Flow Sensing and Control via POD and LSE Base Low-Dimensional Tools, 2002 ASME Fluids Engineering Division, Summer Meeting, Montreal, ASME Paper FEDSM 2002-31416, To appear, J. Fluids Eng., March 2004, hereby incorporated by reference, that demonstrates in the “ActiveWing” facility that such methods could be used for state estimation from wall pressure alone. The background of the present invention is also premised on Glauser, M., Young, M., Higuchi, H., Tinney, C. and Carlson, 2004, H. POD Based Experimental Flow Control on a NACA-4412 Airfoil (Invited), 42nd AIAA Aerospace Sciences Meeting and Exhibit—AIAA 2004-0575, hereby incorporated by reference.
The ActiveWing facility consists of a backward ramp and a variable geometry flap mounted above the ramp. The adverse pressure gradient can be altered by varying the position of the flap. As the flap angle is raised from a position parallel to the ramp to fully raised, the flow transitions from a channel flow, through a point of incipient separation, and finally to a separated flow. The ActiveWing flow study was performed to investigate the benefits of developing low-dimensional descriptions using a general basis set which includes information from all flow states, as opposed to a basis set optimized for a particular flap angle. Taylor and Glauser found that a 5 mode POD model predicts incipient separation effectively. In addition a 5 mode POD/mLSE model for the system state estimate (from wall pressure alone) captures the general features of the velocity field/POD expansion coefficients, which is key for implementation of closed-loop control. While the ActiveWing facility is essentially an internal flow, the present invention encompasses methods that are applicable to the external flow over the NACA 4412 airfoil.
Siegel, S., Cohen, K. and McLaughlin, T., 2003, Feedback Control of a Circular Cylinder Wake in Experiment and Simulation (Invited), 33rd AIAA Fluid Dynamics Conference and Exhibit—AIAA 2003-3569, hereby incorporated by reference, demonstrated POD based feedback control for the external flow over a circular cylinder wake with excellent results indicating that such methods indeed work for external flows as well. This report, however, used inflow measurements and not the practical surface measurements of the present invention. The recent applications of POD/mLSE by Schmit, R. and Glauser, M., 2003, Low Dimensional Tools for Flow-Structure Interaction Problems: Application to Micro Air Vehicles, 41st AIAA Aerospace Sciences Meeting and Exhibit—AIAA 2003-0626, and Schmit, R. and Glauser, M., 2004, Improvements in Low Dimensional Tools for Flow-Structure Interaction Problems: Using Global POD, 42nd AIAA Aerospace Sciences Meeting and Exhibit—AIAA 2004-0889, hereby incorporated by reference, to a Micro Air Vehicle wing wake flow demonstrate the utility of using the mLSE method for external flows as well. These trials were able to estimate with reasonable fidelity the velocity field in the wake just from dynamic strain gages mounted on the flexible wing structure.
Glauser, M, Young, M, Higuchi, H., Tinney, C. and Carlson, H.POD Based Experimental Flow Control on a NACA-4412 Airfoil (Invited), 42nd AIAA Aerospace Sciences Meeting and Exhibit—AIAA 2004-0575, hereby incorporated by reference, showed that an estimation method works well for the NACA 4412 foil, and thus provide a key foundation for the present invention.
In 1967, Lumley, J. L., The structure of inhomogeneous turbulent flows, Atm. Turb. and Radio Wave Prop., Nauka, Moscow and Toulouse, France, Yaglom and Tatarsky eds., pp. 166-178 (1967), hereby incorporated by reference, proposed POD as an unbiased technique for studying coherent structures in turbulent flows. POD is a logical way to build basis functions which emphasize the energetic features of the flow (Holmes, P. J., Lumley, J. L., Berkooz, G., Mattingly, J. C. & Wittenberg, R. W., Low-Dimensional Models of Coherent Structures in Turbulence, Physics Reports, v. 287, pp. 337-384 (1997), hereby incorporated by reference). This results in a small number of the structures containing a large percentage of the system dynamics.
POD is a straightforward mathematical approach based on the Karhunen-Loeve expansion. It is used to decompose the velocity field 1 of 9 into a finite number of empirical functions, which can be used to ascertain a subspace where a model can be constructed by projecting the governing equations on it (Holmes, P. J., Lumley, J. L. & Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press (1996), hereby incorporated by reference). These functions, φ, are linearly independent and form a basis set chosen to maximize the mean square projection of the velocity field. The eigenfunctions are obtained from the following integral eigenvalue problem:∫Rij({right arrow over (x)},{right arrow over (x)}′)φj(n)({right arrow over (x)}′),d{right arrow over (x)}′=λ(n)φ0({right arrow over (x)}).  (1)
The kernel of equation 1 is the ensemble averaged two-point spatial velocity correlation tensor, Rij({right arrow over (x)},{right arrow over (x)}′), which is defined asRij({right arrow over (x)},{right arrow over (x)}′)=ūi({right arrow over (x)},to)uj({right arrow over (x)}′,to)  (2)
where to is a given time snapshot. The eigenfunctions of equation (1) give the optimal basis, and are termed empirical eigenfunctions since they are derived from the ensemble of the observations. The Hilbert-Schmidt theory ensures that if the random field occurs over a finite domain, an infinite number of orthonormal solutions can be used to express the original random velocity field, ui,
                                          u            i                    ⁡                      (                                          x                →                            ,              t                        )                          =                              ∑                          n              =              1                        ∞                    ⁢                                                    a                n                            ⁡                              (                t                )                                      ⁢                                          ϕ                i                                  (                  n                  )                                            ⁡                              (                                  x                  →                                )                                                                        (        3        )            
where the coefficients are defined as,αn(t)=∫Dui({right arrow over (x)},t)φi(n)−({right arrow over (x)})d{right arrow over (x)}  (4)
In 1977, Adrian, R. j., On the role of conditional average in turbulence theory, Turbulence in Liquids: Proceedings of the Fourth Biennial Symposium on Turbulence in Liquids, Science Press, Zakin, J. & Patterson, G., eds., pp. 323-332 (1977), hereby incorporated by reference, proposed the application of stochastic estimation to instantaneous data. Adrian recognized that the statistical information contained within the two-point correlation tensor, Rij, could be combined with instantaneous information to form a technique for estimating the flow field. Cole, D. R., Glauser, M. N. & Guezennec, Y. G., An Application of Stochastic Estimation to the Jet Mixing, Layer. Phys. Fluids, 4(1), pp. 192-194 (1991), hereby incorporated by reference, demonstrated this in the axisymmetric jet shear layer where they successfully estimated the velocity radially across the jet shear layer using information from only a few radial locations. Bonnet, J. P., Cole, D. R., Delville, J., Glauser, M. N. & Ukeiley, L. S., Stochastic estimation and proper orthogonal decomposition: Complementary techniques for identifying structure, Experiments in Fluids. 17 pp. 307-314 (1994), hereby incorporated by reference, expanded on the work of Adrian (1977) and Cole et al. (1991) to form the complementary technique which combines the POD and LSE to obtain the time dependent POD expansion coefficients from instantaneous velocity data on course hot wire grids.
Taylor and Glauser (2002, 2004) further expanded these methods and demonstrated how instantaneous wall pressure measurements could be used to construct an accurate representation of the instantaneous velocity field from wall pressure alone (i.e., the modified complementary technique or modified linear stochastic estimation (mLSE)). This approach can be applied to the POD using either the “conditional” or “global” POD eigenfunctions described above. Boree, J., Extended proper orthogonal decomposition: A tool to analyze correlated events in turbulent flows, Experiments in Fluids 35, pp. 188-192 (2003) and Fogleman, M., Lumley, J. L., Rempfer, D. and Haworth, D., Analysis of tumble breakdown in ic engine flows, To appear Physics of Fluids (2004), hereby incorporated by reference, apply a similar approach to engine cylinder flow, but the approach has not been used to determine the flow velocity over an airfoil, nor has it been used to control airfoil actuators.
3. Objects and Advantages
It is a principal object and advantage of the present invention to provide a method for using Proper Orthogonal Decomposition and Modified Linear Stochastic Estimation to determine the flow velocity over an airfoil.
It is an additional object and advantage of the present invention to provide a method for feedback control over airfoil actuator using Proper Orthogonal Decomposition and Modified Linear Stochastic Estimation to determine the flow velocity and a feedback loop.
Other objects and advantages of the present invention will in part be obvious, and in part appear hereinafter.